52.9.5 problem 5
Internal
problem
ID
[8383]
Book
:
DIFFERENTIAL
EQUATIONS
with
Boundary
Value
Problems.
DENNIS
G.
ZILL,
WARREN
S.
WRIGHT,
MICHAEL
R.
CULLEN.
Brooks/Cole.
Boston,
MA.
2013.
8th
edition.
Section
:
CHAPTER
8
SYSTEMS
OF
LINEAR
FIRST-ORDER
DIFFERENTIAL
EQUATIONS.
EXERCISES
8.1.
Page
332
Problem
number
:
5
Date
solved
:
Monday, January 27, 2025 at 03:50:13 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )-y+z \left (t \right )+t -1\\ y^{\prime }&=2 x \left (t \right )+y-z \left (t \right )-3 t^{2}\\ z^{\prime }\left (t \right )&=x \left (t \right )+y+z \left (t \right )+t^{2}-t +2 \end{align*}
✓ Solution by Maple
Time used: 0.142 (sec). Leaf size: 171
dsolve([diff(x(t),t)=x(t)-y(t)+z(t)+t-1,diff(y(t),t)=2*x(t)+y(t)-z(t)-3*t^2,diff(z(t),t)=x(t)+y(t)+z(t)+t^2-t+2],singsol=all)
\begin{align*}
x \left (t \right ) &= t^{2}-\frac {1}{6}+c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )+c_3 \,{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \\
y &= -\frac {t^{2}}{2}-\frac {3 t}{2}+\frac {c_{1} {\mathrm e}^{2 t}}{2}-\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )}{2}+\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{\frac {t}{2}} \sqrt {11}\, \cos \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {7}{4} \\
z \left (t \right ) &= -\frac {t}{2}+\frac {3 c_{1} {\mathrm e}^{2 t}}{2}-c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )-c_3 \,{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right )-\frac {3 t^{2}}{2}-\frac {7}{12} \\
\end{align*}
✓ Solution by Mathematica
Time used: 12.552 (sec). Leaf size: 304
DSolve[{D[x[t],t]==x[t]-y[t]+z[t]+t-1,D[y[t],t]==2*x[t]+y[t]-z[t]-3*t^2,D[z[t],t]==x[t]+y[t]+z[t]+t^2-t+2},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to t^2+\frac {2}{5} c_1 e^{2 t}+\frac {2}{5} c_3 e^{2 t}+\frac {1}{5} (3 c_1-2 c_3) e^{t/2} \cos \left (\frac {\sqrt {11} t}{2}\right )-\frac {(c_1+10 c_2-4 c_3) e^{t/2} \sin \left (\frac {\sqrt {11} t}{2}\right )}{5 \sqrt {11}}-\frac {1}{6} \\
y(t)\to \frac {1}{220} \left (-11 \left (10 t^2+30 t-4 (c_1+c_3) e^{2 t}+35\right )-44 (c_1-5 c_2+c_3) e^{t/2} \cos \left (\frac {\sqrt {11} t}{2}\right )+4 \sqrt {11} (17 c_1+5 c_2-13 c_3) e^{t/2} \sin \left (\frac {\sqrt {11} t}{2}\right )\right ) \\
z(t)\to -\frac {3 t^2}{2}-\frac {t}{2}+\frac {3}{5} (c_1+c_3) e^{2 t}-\frac {1}{5} (3 c_1-2 c_3) e^{t/2} \cos \left (\frac {\sqrt {11} t}{2}\right )+\frac {(c_1+10 c_2-4 c_3) e^{t/2} \sin \left (\frac {\sqrt {11} t}{2}\right )}{5 \sqrt {11}}-\frac {7}{12} \\
\end{align*}